The sensors of angular position (tiltmeters) are widely used not only for scientific applications but also for industrial and civil applications. Typical applications of the sensors of angular position, wherein a highest sensitivity and accuracy of the measurements is requested in the low frequencies band, are, for example, both in the scientific domain (monitoring of the ground elastic deformations, terrestrial tides, etc.) both in the civil and industrial domains (monitoring of dykes, tunnels, bridges, mines, monuments and buildings, even in relation to the evaluation of the seismic risk).
The current classification of the existing instruments is effected on the basis of the length of the base of the instrument: long-base and short-base sensors. The angular short-base sensors have the great advantage of being easily installable, but have the disadvantage of being more sensible to the non-homogeneities and the local perturbations, as well as the local perturbations of the environmental and/or meteorological type. The long-base angular sensors, although obviously more complex to install, more expensive and more sensible to the meteorological-type perturbations, provide more accurate and stable measurements of angular displacements, provided that, however, the sites are suitable and the fixing to ground of the sensors are appropriately designed [14-19].
A typical example of angular long-base sensor is the “water-tube tiltmeter” [14], whose sensitivity depends on the horizontal distance between two water containers: the larger is the distance, the greater is the sensitivity. The other side of the coin is the fact that a liquid angular displacement sensor (tiltmeter) detects only a mean value of the tilts on a large scale. Moreover, these instruments are largely influenced by the ambient conditions (e.g. liquid density variations due to temperature variations, surface tensions acting on the walls of the liquid containers, asymmetry of the containers and the transducers), which make difficult the achieving of high sensitivities.
Typical examples of short-base sensors are the pendula, the diamagnetic and bubble angular sensors [17-19]. Even such sensors present problems of sensitivity, stability along time, decoupling of the angular position signal from the other degrees of freedom, decoupling with the ambient noises that could be remarkably reduced in the assumption of utilizing architectures of the starting sensor which unite the sensitivity of the long-base sensors, the compactness of the short-base sensors, with the ensuing smaller sensitivity to the ambient noises and an effective decoupling between the degrees of freedom.
A possible architecture is that of the folded pendulum, as a matter of fact a low-frequency oscillating system based on the Watt-linkage [1], realized also in the monolithic form with mechanical working by milling and electro-erosion, both in the classical experimental version with joints in tension [6-8], and in the version with a pair of joints in compression [9], and in the new vertical version [10]. Such an architecture guarantees the realization of sensors characterized by a wide measurement band coupled with a highest sensitivity in the band of low frequencies, full dimensional scalability of the sensor, full tuneability of the resonance frequency, high mechanical quality factors, reduced problems of coupling between the various degrees of freedom and reduced sensitivity to the ambient noises, also as a consequence of an efficient system of reading the output signal, the system being based on optic-electronic methods, such as, for example, optical levers or laser interferometers [7-13].
It is well known, however, that the output signal obtained by reading the relative motion of the central mass with respect to the pendulum support, as obtained in all the configurations realized as of yet and described in literature [1-13], is a combination of the linear displacement (component of the relative motion of the central mass with respect to the pendulum support due to forces acting parallel to the base of the same support) and the angular displacement (tilt) of the support base.
Indeed, making reference to FIG. 1, all the existing techniques and methodologies relevant to the folded pendulum, both for the measurement of horizontal displacements (and/or accelerations), vertical displacements (and/or accelerations) and angular displacements (and/or accelerations) are based on the only direct measurements of the displacement of the central mass (mc) with respect to the support (F). Such a support is rigidly fixed to the surface of which one wishes to measure the linear and/or angular displacements, constituting with it an only rigid block and, therefore, following its displacements in a rigid way. Accordingly, the measurement of the displacement of the central mass with respect to the support is a direct measurement of the displacement of the central mass (test mass) with respect to the surface. The problem that the current systems do not allow to solve is that the signals in input to each folded pendulum are in theory three (horizontal, vertical and angular displacements) but the reading system acquires only a combination of them [2-5]. The system is clearly undetermined: it is not mathematically possible in any manner to obtain, starting from an only reading signal, the three input signals, unless one makes very severe assumptions on the latter or one modifies the performances of the instrument in such a way to maximize the signal-to-noise ratio of a signal with respect to the others.
Only in some particular cases it has been possible to solve the problem in a partial way, for example in the case wherein the linear and angular displacements are ideally separated in band, or under the assumption to use concurrently also an independent sensor (tiltmeter) of only angular displacement having comparable sensitivity (with evident technical problems connected to different sensitivities, stability and calibration criticalities).
Indeed, in the article of Takamori et al. [5], the authors utilize the folded pendulum in a classical way by fixing the support rigidly to the surface whose angular displacements are to be measured along time, and measure the displacement of the central mass with respect to the support. In particular, they highlight the already well known equivalence of the folded pendulum with respect to a classical pendulum with equivalent length, leq (equation 1 in the article), and period T0 (equation 2 in the article), fixed in the suspension point to the surface whose angular displacement is to be measured.
As described in the article, and well known in the literature, by keeping the same notation utilized there, a rotation of an angle Δϑ of the reference plane, to which the pendulum is fixed, around an axis perpendicular to the plane on which occurs the motion of the folded pendulum generates a measurable quantity, i.e. a displacement of the central mass relatively to the support, Δd, that can be expressed by the trigonometric relationship Δd=leq·Δϑ for sufficiently small Δϑ.
It is important to stress that the period utilized in the paper of Takamori, that is the inverse ratio of the natural resonance frequency of the folded pendulum, is considered constant independently from any possible time variation of the inclination angle of the support with respect to the plane perpendicular to the gravity force vector, {right arrow over (g)}, and, therefore, the result of the measurement is independent of the variation of such a period.
It is, however, as much known and scientifically well-established that the measurement of the relative displacement of the central mass with respect to the support, fixed to ground, is not per se sufficient for the carrying-out of a univocal measurement of angular displacement. Indeed, it has been scientifically demonstrated also in the article of Takamori, that the measurement of the relative displacement between central mass and support is a combination of angular displacement and linear displacements (coupling between the horizontal motion of the plane and the vertical motion of the plane as referred to the direction of the gravitational acceleration vector {right arrow over (g)}). Moreover, it is not possible by means of a single measurement to decouple a linear motion from an angular one, and subsequently from the linear motion decouple the horizontal and vertical local components. This, besides being easily derivable from the papers is mathematically unexceptionable. However, whilst it is not possible to have a static displacement of the central mass (test mass) (except for the case of a constant acceleration), it is instead possible to have a static angular displacement, as a consequence of the presence of the gravitational acceleration. It is here stressed that the foregoing is valid only in the static case: dynamical angular displacements cannot be instead in no way distinguished from dynamical linear displacements.
As a demonstration of the foregoing, the authors of [5] are forced to take into account this problem. Indeed, they discuss the consequences of a micro-seism on their measurement. The micro-seism they are speaking of, close to the resonance frequency of the pendulum, is for them a great problem for two reasons:
1. If the linear micro-seism is large, it does not allow them to measure the angular displacement in a correct way, since the output signal measured by them is a combination of the signals of linear and angular displacement.
2. In particular, if the micro-seism is close to the resonance frequency, this signal makes the pendulum oscillate, creating problems not only of de-coupling, but also of dynamics, which in the sensor developed by them is relatively small (see the figures of the article).
The solution implemented by Takamori et al. is to damp the oscillations of the sensor to the end of annul any possibility of triggering oscillations in the sensor (by reducing the quality factor to Q=1 by means of magnetic dampers) and carry out only measurements of very long period, in the frequency band wherein one presumes that the linear seismic signal is sufficiently small to be considered small with respect to the signal of angular displacement.
The limitations of that method, as well as the complexity of its application, are evident.